Long-time Asymptotics for Nonlinear Growth-fragmentation Equations

Details Long-time Asymptotics for Nonlinear Growth-fragmentation Equations Long-time Asymptotics for Nonlinear Growth-fragmentation Equations

2011-02-14

arxiv.org/abs/1102.2871v1

Mathematics

Analysis of PDEs

Scientific paper

We are interested in the long-time asymptotic behavior of growth-fragmentation equations with a nonlinear growth term. We present examples for which we can prove either the convergence to a steady state or conversely the existence of periodic solutions. Thanks the General Relative Entropy method applied to well chosen self-similar solutions, we show that the equation can "asymptotically" be reduced to a system of ODEs. Then stability results are proved by using a Lyapunov functional, and existence of periodic solutions are proved thanks to the Poincar\'e-Bendixon theorem or by Hopf bifurcation.

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